I need to snap a the face of an object without changing the relationship of the origin point to that object
A number of ways of doing this, but here's one that should preserve all the relationships in your origami. It looks from your picture as if all the folding parts of the model are separate objects, with their origins on the folds, all parented to a static part, with its origin off-plane.
In brief: parent the existing origami-hierarchy to another object aligned to, and in its plane. Then snap the new parent.
Not so brief, and assuming the worst case, with the origami and the target plane both skew to the world and their own object-axes:


With the T &gt; 3D Cursor tool active, and its N &gt; Tool tab options set to 'Surface Project' and Orientation 'Geometry', drag the cursor to somewhere on the static section of your origami.
-ShiftACreate an Empty there, and in its adjustment panel, set its 'Align' to '3D Cursor'
CtrlP parent the static section to the Empty.



If the plane you are aiming at is skew to the world, create a Transform Orientation from its face in Edit Mode by selecting it and clicking the little '+' in the Header's Transform Orientation dropdown. (You won't need to, if the target plane is in World XY.)
With the Empty selected, Header &gt; Object Menu &gt; Transform &gt; Align to Transform Orientation, with the newly created orientation selected. Your origami should now be aligned to the plane.
Snap the Empty to the plane, any way you choose
AltP un-parent your origami from the Empty, with 'Clear and Keep Transformation'.

This really is the worst case: if your origami's local axes are aligned to their planes, or the target plane is aligned to the world, you can skip some of these steps.